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Decidable theories of pseudo-\(p\)-adic closed fields. (English. Russian original) Zbl 0717.12005
Algebra Logic 28, No. 6, 421-438 (1989); translation from Algebra Logika 28, No. 6, 643-669 (1989).
Let \(K\) a field and \(\Gamma\) an ordered Abelian group. A valuation \(\phi: K\to \Gamma \cup \{\infty \}\) is called \(p\)-valuation if \(\text{char}(K)=0\), \(\phi(p)\) is the least element of \(\Gamma\) greater than zero, and the residue class field is the \(p\)-element field. A field \(K\) is called P\(p\)C-field if \(\text{char}(K)=0\) and \(K\) is existentially closed in every regular totally \(p\)-adic extension of \(K\). We recall that the theory of the class of P\(p\)C-fields is undecidable. The author proves that the theory of maximal P\(p\)C-fields and the theories of the classes of P\(p\)C-fields with finitely generated absolute Galois groups are decidable.

12L05 Decidability and field theory
03B25 Decidability of theories and sets of sentences
12J12 Formally \(p\)-adic fields
Full Text: DOI
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